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## Discrete Random Variables

How many heartbeats do you have each minute? How many points will your favorite team score in their game tonight? These any many other real-world values can be modeled by discrete random variables.

# Joint Probability Distribution

The joint probability distribution of discrete random variables $$X$$ and $$Y$$ is given as follows: $\begin{matrix} P(x,y) & x=0 & x=1 & x=2 \\ y=0 & 0.32 & 0.16 & 0.32 \\ y=1 & 0.08 & 0.04 & 0.08 \end{matrix}$ Are $$X$$ and $$Y$$ independent of each other?

Today, Joe's class took a math quiz consisting of two problems. For a randomly selected student, let $$X$$ be the points earned on the first question and $$Y$$ be the points earned on the second question. The joint probability distribution of $$X$$ and $$Y$$ is given in the following table:

$\begin{matrix} & X=0 & X=5 & X=10 & X=15 \\ Y=0 & 0.05 & 0.05 & 0 & 0.07 \\ Y=5 & 0 & 0 & 0.18 & 0.07 \\ Y=10 & 0 & 0.11 & 0.36 & 0.11 \end{matrix}$

What is the expected points that a randomly selected student got?

If the joint probability distribution of $$X$$ and $$Y$$ is given by as follows: $\begin{matrix} P(x,y) & x=0 & x=1 & x=2 & x=3 \\ y=0 & 0 & 0.05 & 0 & 0.04 \\ y=1 & 0.04 & 0.41 & 0.05 & 0.05 \\ y=2 & 0.05 & 0.04 & 0.04 & 0.05 \\ y=3 & 0.05 & 0.04 & 0.05 & 0.04 \end{matrix}$ what is $$E[X+Y]?$$

If the joint probability $$P(x,y)$$ of $$X$$ and $$Y$$ is given by as follows: $\begin{matrix} P(x,y) & x=0 & x=1 & x=2 & x=3 & x=4 \\ y=0 & 0 & 0.03 & 0.03 & 0.05 & 0.04 \\ y=1 & 0.29 & 0 & 0.06 & 0.05 & 0.04 \\ y=2 & 0 & 0.06 & 0.03 & 0.05 & 0.05 \\ y=3 & 0.05 & 0.05 & 0.05 & 0.03 & 0.04 \end{matrix}$ what is $$E[XY]?$$

Today, Joe's class took a math quiz consisting of two problems. For a randomly selected student, let $$X$$ be the points earned on the first question and $$Y$$ be the points earned on the second question. The joint probability distribution of $$X$$ and $$Y$$ is given in the following table:

$\begin{matrix} & X=0 & X=5 & X=10 & X=15 \\ Y=0 & 0.07 & 0.07 & 0 & 0.07 \\ Y=5 & 0 & 0 & 0.18 & 0.07 \\ Y=10 & 0 & 0.11 & 0.32 & 0.11 \end{matrix}$

The students who have earned a total of less than $$15$$ points will be taking a makeup test. What is the probability that a randomly chosen student will be taking a makeup test?

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